Both are great rounds, and you cannot go wrong with either. I have two Weatherby's and like them, so I am not a Wby basher. But that said, I would go with the 338RUM. Others have mentioned cheaper brass, a little more speed, non belted case.........all are positives. I have a 338RUM in an HS-Precision and it shoots great.
The 338 Edge is also a good improvement over the RUM but you have to fire form brass. Not a real big deal, but worth considering. It will get you some more speed though and that is always a good thing [img]/ubbthreads/images/graemlins/grin.gif[/img]
Last, put a muzzle brake
on that thing!! You will not regret it. For long range shooting you will need to be looking at the heavier bullets 225's and 250's. (I like 250grn Sierra Match Kings myself) With those larger bullets and a full load of powder you will get a real wake-up call if you don't have a brake [img]/ubbthreads/images/graemlins/shocked.gif[/img]